J. Phys. Chem. A 2004, 108, 4405-4411
4405
Ultraviolet Spectroscopy of Large Water Clusters: Model and Calculations for (H2O)n, for n ) 8, 11, 20, 40, and 50 Y. Miller,† E. Fredj,‡ J. N. Harvey,§ and R. B. Gerber*,†,# Department of Physical Chemistry and The Fritz Haber Research Center, The Hebrew UniVersity, Jerusalem 91904, Israel, Computer Science Department, Jerusalem College of TechnologysMachon LeV, Jerusalem 91160, Israel, School of Chemistry, UniVersity of Bristol, Bristol BS8 1TS, England, and Department of Chemistry, UniVersity of California, IrVine, California 92697-2025 ReceiVed: May 30, 2003; In Final Form: January 27, 2004
The UV absorption spectra of neat water clusters (H2O)n of sizes in the range of n ) 8-50 are computed. The simple model used for the excited states includes the dependence of the excitonic interactions on both the intermolecular and intramolecular coordinates. For a cluster (H2O)n, n excitonic potential energy surfaces are computed for geometries in the Franck-Condon region. The Coker-Watts potential is used to describe the interactions in the electronic ground state, and molecular dynamics simulations are performed to sample geometries for the classical Franck-Condon calculations. There are numerous crossings of different excitonic potential surfaces for (H2O)n in the range of the geometries sampled. The main findings are (i) the main absorption peak of (H2O)n shifts to the blue and increases in width as the cluster size n is increased; (ii) the widths of the absorption bands increase with temperature, e.g., for (H2O)20, the width is 1.2 eV at 80 K and 1.6 eV at 220 K; (iii) several well-resolved peaks within the absorption band are found for some of the systems at certain temperatures, and in such cases, each of the peaks generally results from absorption into different excitonic states; (iv) although the absorption peaks are strongly shifted to the blue, with respect to the (H2O) monomer, for some cluster sizes, a weak absorption tail to the red side is also observed as the temperature increases.
I. Introduction Water clusters of different size ranges have been extensively studied recently, both experimentally and theoretically.1 Much progress has been made on the characterization and understanding of many properties of these systems. This is the case for the vibrational spectroscopy of these clusters, which has blossomed into a broad area of research, with the introduction of novel experimental techniques and the development of improved theoretical models. It is especially true for relatively small clusters, with rigorous calculations and simulations.2-13 The situation is quite different for the electronic spectroscopy of neat water clusters. Relatively little seems to be known on this topic, although it is of potential importance for the understanding of a range of phenomena. For example, electronic spectroscopy is essential for the study of photochemical processes, and water clusters offer a very interesting framework for the exploration of photochemistry in hydrogen-bonded networks. The electronic spectroscopy of finite water clusters is very useful for understanding the electronic properties of liquid water in bulk, and those of ices. The excited electronic states of the water molecule, and the related photodissociation dynamics in the gas phase, have been studied in detail. The UV absorption cross sections of the three lowest bands (A ˜ 1B1, B ˜ 1A1, and C ˜ 1B1, respectively) were measured14 and ab initio calculations of the excited electronic states and the transition moments were reported.15 * Author to whom correspondence should be addressed. E-mail address:
[email protected]. † The Hebrew University. ‡ Jerusalem College of TechnologysMachon Lev. § University of Bristol. # University of California, Irvine.
There is a wealth of experimental data on vibrational and rotational state distributions on the electronic states of the photofragments and on vibrational and rotational product state distributions.16-19 Good potential energy surfaces are available from ab initio calculations.20,21 Finally, extensive calculations on the photodissociation dynamics have been reported.9,22 Such detailed data are not available for water clusters. Ab initio calculations on the lowest excited singlet state of small water clusters have been performed by van Hemert and van der Avoird,23 Sosa et al.,24 and Sobolewski and Domcke.25 Useful hints on the UV spectroscopy of water clusters can be inferred from experimental data on spectroscopy and photodissociation of liquid water and of ice in bulk.26-28 Model calculations of electronic excitation spectra of water in bulk are also available.29-31 The present study continues and expands on a previous theoretical investigation by Harvey, Jung, and Gerber.32 In that paper, the UV absorption spectra of small water clusters (H2O)n, where n ) 2-6, were calculated from a theoretical model of excited electronic states. The calculations were conducted for water clusters initially at 0 K, i.e., in the vibrational ground state. In the present paper, we use the same simple model for the excited-state potential to explore the UV absorption spectrum of water clusters (H2O)n of various sizes (up to n ) 50). In addition, we study the temperature dependence of the absorption bands. In some of the calculations, the clusters are in a “solidlike” state, whereas in other calculations, their state is essentially in the liquid regime. The objective of these calculations is to serve as a guide for future experiments. The structure of the article is as follows. The theoretical model used in the calculations is presented in Section II. The results of the simulations are described and discussed in Section III. Concluding remarks are presented in Section IV.
10.1021/jp030678b CCC: $27.50 © 2004 American Chemical Society Published on Web 04/28/2004
4406 J. Phys. Chem. A, Vol. 108, No. 20, 2004 II. Model and Methods To determine trends in the dependence of the spectroscopy on cluster size, we performed calculations for the cluster sizes n ) 8, 11, 20, 40, and 50. The UV spectra of smaller water clusters (n ) 2-6) were studied theoretically by Harvey et al.32 That study was limited to clusters initially in their vibrational ground state. In the present paper, we investigate the temperature dependence of the electronic absorption spectroscopy for clusters in the temperature range from T ) 50 K to T ) 300 K. This range covers both solid-like and liquid-like states of the clusters. A. Simulations of the Initial State. Calculation of the Franck-Condon factors requires sampling of the configurations of the system at a given temperature. We assume that, for the temperatures considered here (T g 50 K), classical dynamics can be used for the purpose of sampling configurations. Obviously, the temperatures considered are very low, compared with the zero-point energies of the stiff intramolecular vibrations. However, the distribution of cluster configurations is mostly determined by the soft intermolecular motions of the water molecules, and for these low-frequency modes, classical dynamics should be a reasonable approximation at and above 50 K. However, the intramolecular vibrations of the monomers were included in the modeling. We used the Coker-Watts potentials33 for the intramolecular force field and for interactions between water molecules in the electronic ground state. The CokerWatts model is not the most-accurate potential currently available for the interaction between vibrationally flexible water molecules; for example, it does not include three-body and higher-order effects. However, the objectives of this study are only semiquantitative, and we estimate that, for this purpose, the accuracy of the Coker-Watts model is sufficient. The Coker-Watts modes were adopted here for their computational convenience. To sample configurations for a cluster at a given temperature, molecular dynamics (MD) simulations were performed. The trajectories were computed using the standard Verlet algorithm and propagated in time to equilibration.34 The standard MD tests have shown that equilibrium was attained within time scales on the order of 20 ps. (The equilibration time differs for each system.) The propagation of the trajectories was pursued beyond equilibration, and configurations were sampled at random times. The set of these configurations represents a sampling of the equilibrium state of the cluster, and photoexcitation is assumed to occur from this initial state. Each simulation involved the calculation of a single, sufficiently long trajectory for a fixed, conserved total energy. The temperature of such a fixed-energy system obviously fluctuates in time; however, the fluctuations were computed and determined to be very small (∆T j1 K, typically). Thus, the temperature was practically well-defined in the simulations. There are large-amplitude motions in the clusters at the temperatures studied. Nevertheless, the minimum-energy structure is often useful for interpretation and analysis. Thus, a search for the most-stable structure was conducted for each cluster. In several cases, other low-energy isomers were computed, because knowledge of these structures can also be useful in the interpretation of the spectra. The calculation of the structures was performed by the simulated annealing method. There is no certainty in these calculations that, indeed, the global minimum structure would be determined. Given the extensive searches that we pursued, we assume that the lowest-energy isomer was determined in most cases. Except for the lowest temperatures used, several local minimum structures are sampled by the trajectories; however, these structures are not necessarily very
Miller et al. different geometrically. The low-temperature simulations seem to be confined to the vicinity of the global minimum structure. B. Modeling of the Excited Electronic States. The excitation modeled in the present study corresponds to the A ˜ 1B1 band in the case of the monomer, which is the lowest excitation band. Consider the cluster (H2O)n for a first configuration of the atoms. If interactions between the water molecules are ignored, there are n nearly degenerate excited energies in the band for that configuration, each corresponding to an excitation of a different water molecule in the cluster. The excitation energies of different water molecules can differ, if their internal geometries are slightly different, representing different vibrational displacements from equilibrium. The n excited states are exactly degenerate for configurations where all molecules in the cluster have exactly the same vibrational displacements. When the effects of interactions between the water molecules are incorporated, including interaction between the excited monomer and the other molecules, the result will be the formation of n excitonic states from the n localized excited states of the different H2O molecules. When treating the excitonic states, we followed the model of Harvey et al.32 previously used for (H2O)n, for n ) 2-6. We constructed the matrix Hamiltonian in the representation of the basis of the uncoupled water molecules. The ith basis function can be written as (ex) Φ(0) i ) Φi (ri,Ri)
φj(g)(rj,Rj) ∏ j*i
(1)
where ri denotes the electronic coordinates of the ith water molecule, and Ri is the nuclear configuration of that same molecule. Φ(ex) is the excited state of the ith water molecule, i and Φ(g) denotes the ground electronic state of molecule j. j Specifically, the excited state of Φ(ex) ˜ 1B1 state of i (ri,Ri) is the A water. Studies of the water monomer indicated that the 1B1 state is obtained by promoting an electron from the lone-pair 1b1 orbital of the ground state into the part-Rydberg, part-antibonding 4al orbital.15,20 The diagonal elements Hii of the Hamiltonian model in the representation of the basis described in eq 1 are given by32
Hii ) Vi* +
Vj + ∑ Vj,k ∑j Vi*,j + ∑ j*i j
